Why can't I use loop cuts after beveling?
I tried that with Blender 2.81 and a beveled cube. It works. So it seems that the bevel is not the reason for the problem.
Check the geometry: Go into edit mode an "Merge Vertice by distance" (v2.80) or "Merge" (CTRL-V in edit mode). Ensure that the normals of the faces are pointing outside. If this does not help there is a workaround: In edit mode select all edges in question and subdivide instead of loop cut.